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152                     ANALYTICAL MECHANICS
ated by revolving about the z-axis the area bounded by y* = 2 px and x = a. Find the center of mass of the remaining solid if the paraboloid and the cone have the same base and vertex.
120. Definition of Moment of Inertia.  The moment of inertia of a body about an axis equals the sum of the products of the masses of the particles of the body by the square of their distances from the axis.* Thus if dm denotes an element of mass of the body and r its distance from the axis then the following is the analytical statement of the definition of moment of inertia:
/= PVdm.                              (II)
The integration which is involved in equation (II) is often simplified by a proper choice of the element of mass. The choice depends upon the bounding surfaces of the body and the position of the axis; therefore there is no general rule by which the'most convenient element of mass may be selected. There is one important point, however, which the student should always keep in mind in -selecting the element of mass, namely, the distances of the various parts of the element of mass from the axis must not differ by more than infinitesimal lengths.
1. Find the moment of inertia of a rectangular lamina about one of its .sides.
Suppose the lamina to lie in the xy-plane. Further suppose the side with respect to which the moment of inertia is to be found to lie in the rr-axis. Then the most convenient element of mass is a strip which is parallel to the aj-axis.
Let a be the length (Fig. 84), b the                  FlQ' 84"
width, and cr the mass per unit area of the lamina, then
dm  era dy.
* For a physical definition of moment of inertia and its meaning see p. 220.